Pathwise upper semi-continuity of random pullback attractors along the time axis

By Hongyong Cui, Peter E. Kloeden, Fuke Wu. Physica A: Statistical Mechanics and its Applications, 2018

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Abstract The pullback attractor of a non-autonomous random dynamical system is a time-indexed family of random sets, typically having the form { A t ( ⋅ ) } t ∈ R with each A t ( ⋅ ) a random set. This paper is concerned with the nature of such time-dependence. It is shown that the upper semi-continuity of the mapping t ↦ A t ( ω ) for each ω fixed has an equivalence relationship with the uniform compactness of the local union ∪ s ∈ I A s ( ω ) , where I ⊂ R is compact. Applied to a semi-linear degenerate parabolic equation with additive noise and a wave equation with multiplicative noise we show that, in order to prove the above locally uniform compactness and upper semi-continuity, no additional conditions are required, in which sense the two properties appear to be general properties satisfied by a large number of real models.

Subfields: Computational Theory and Mathematics, Computer Networks and Communications, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/be24d0454d7b81f0a25434bbb53c1c7a67acafd4